F-signature of pairs and the asymptotic behavior of Frobenius splittings

Abstract

We generalize $F$-signature to pairs $(R,D)$ where $D$ is a Cartier subalgebra on $R$ as defined by the first two authors. In particular, we show the existence and positivity of the $F$-signature for any strongly $F$-regular pair. In one application, we answer an open question of I. Aberbach and F. Enescu by showing that the $F$-splitting ratio of an arbitrary $F$-pure local ring is strictly positive. Furthermore, we derive effective methods for computing the $F$-signature and the $F$-splitting ratio in the spirit of the work of R. Fedder.

27 pages, exposition improved, typos corrected, references added. To appear in Advances in Mathematics

References in corpus (3)

• F-thresholds and Bernstein-Sato polynomials
• F-signature exists
• A survey of test ideals

Cited by in corpus (6)

• F-signature exists
• Fundamental groups of $F$-regular singularities via $F$-signature
• Depth of $F$-singularities and base change of relative canonical sheaves
• On the three dimensional minimal model program in positive characteristic
• F-Signature of Affine Toric Varieties
• Relating F-Signature and F-Splitting Ratio of Pairs Using Left-Derivatives